Introduction

The marketing and accounts receivable managers at our company notify us that we have significant exposure to exchange rates. Our functional currency is U.S Dollars (USD), but we operate in the United Kingdom, European Union, and Japan. The exposure to exchange rates hits the gross revenue of out financial lines.

Our cash flow is also affected by the ebb and flow of accounts receivable components of working capital in producing and selling several products. When the exchange rates are volatile, so are our earnings. The goal of this project is to explore the relationships between the different markets to get a better understanding of how our earnings are affected by the exchange markets. This is especially important as we have missed our earnings forecasts for five consecutive quarters.

Part 1

Importing the Data

First, we will load in time series data of the exchange rates for the European, United Kingdom, Chinese, and Japanese markets. We will obtain the csv file from the turing.manhattan.edu website and view the structure and a sample of the exchange rates file.

library(zoo)      #For creating time series objects
library(xts)      #For time series analysis
library(ggplot2)  #For creating graphics
library(plotly)

#The URL for the exchange data data
URL <- "https://turing.manhattan.edu/~wfoote01/finalytics/data/exrates.csv"

#Reading in the exchange rates and omitting the missing data from the 
#url provided by turing.manhattan.edu and keeping the dates as characters
exrates <- na.omit(read.csv(url(URL), stringsAsFactors = F))
#Converting the string dates to actual dates
exrates$DATE <- as.Date(exrates$DATE, "%m/%d/%Y")

#Five columns (date, eur2usd, gbp2usd, cny2usd, jpy2usd)
#the data is daily exchange rates
head(exrates)     #Looking at the data
##         DATE USD.EUR USD.GBP USD.CNY USD.JPY
## 1 2013-01-28  1.3459  1.5686  6.2240   90.73
## 2 2013-01-29  1.3484  1.5751  6.2259   90.65
## 3 2013-01-30  1.3564  1.5793  6.2204   91.05
## 4 2013-01-31  1.3584  1.5856  6.2186   91.28
## 5 2013-02-01  1.3692  1.5744  6.2265   92.54
## 6 2013-02-04  1.3527  1.5737  6.2326   92.57
tail(exrates)     #Looking at the end of the data
##            DATE USD.EUR USD.GBP USD.CNY USD.JPY
## 1248 2018-01-19  1.2238  1.3857  6.3990  110.56
## 1249 2018-01-22  1.2230  1.3944  6.4035  111.15
## 1250 2018-01-23  1.2277  1.3968  6.4000  110.46
## 1251 2018-01-24  1.2390  1.4198  6.3650  109.15
## 1252 2018-01-25  1.2488  1.4264  6.3189  108.70
## 1253 2018-01-26  1.2422  1.4179  6.3199  108.38
str(exrates)      #Viewing the structure of the data
## 'data.frame':    1253 obs. of  5 variables:
##  $ DATE   : Date, format: "2013-01-28" "2013-01-29" ...
##  $ USD.EUR: num  1.35 1.35 1.36 1.36 1.37 ...
##  $ USD.GBP: num  1.57 1.58 1.58 1.59 1.57 ...
##  $ USD.CNY: num  6.22 6.23 6.22 6.22 6.23 ...
##  $ USD.JPY: num  90.7 90.7 91 91.3 92.5 ...
#1253 different instances of exchange rates
summary(exrates)  #From 28 Jan 2013 to 26 Jan 2018
##       DATE               USD.EUR         USD.GBP         USD.CNY     
##  Min.   :2013-01-28   Min.   :1.038   Min.   :1.212   Min.   :6.040  
##  1st Qu.:2014-04-25   1st Qu.:1.107   1st Qu.:1.324   1st Qu.:6.178  
##  Median :2015-07-27   Median :1.158   Median :1.514   Median :6.261  
##  Mean   :2015-07-26   Mean   :1.199   Mean   :1.474   Mean   :6.401  
##  3rd Qu.:2016-10-24   3rd Qu.:1.314   3rd Qu.:1.573   3rd Qu.:6.627  
##  Max.   :2018-01-26   Max.   :1.393   Max.   :1.716   Max.   :6.958  
##     USD.JPY      
##  Min.   : 90.65  
##  1st Qu.:102.14  
##  Median :109.88  
##  Mean   :109.33  
##  3rd Qu.:116.76  
##  Max.   :125.58
# USD to CNY appears to be the most steady

Question 1: Nature of Exchange Rates

Becuase we are interested in how each exchange rate changes over time we will want to look at the percent change in the exchange rate on a daily basis. To calculate the percent change in the exchange rate over time, we will use the difference of logarithms for the sequential data. These calculated numbers will be in the units of percent change.

#Using the properties of the natural log to determine
#the percent change in exchange rates
exrates.r <- diff(log(as.matrix(exrates[,-1]))) * 100
head(exrates.r)   #first 6 days of percent change
##      USD.EUR     USD.GBP     USD.CNY     USD.JPY
## 2  0.1855770  0.41352605  0.03052233 -0.08821260
## 3  0.5915427  0.26629486 -0.08837968  0.44028690
## 4  0.1473405  0.39811737 -0.02894123  0.25228994
## 5  0.7919091 -0.70886373  0.12695761  1.37092779
## 6 -1.2124033 -0.04447127  0.09792040  0.03241316
## 7  0.3100091 -0.54159233 -0.05456676  0.82836254
tail(exrates.r)   #last 6 days of percent change
##          USD.EUR    USD.GBP     USD.CNY    USD.JPY
## 1248  0.00000000 -0.2306640 -0.28869056 -0.2890175
## 1249 -0.06539153  0.6258788  0.07029877  0.5322280
## 1250  0.38356435  0.1719691 -0.05467255 -0.6227176
## 1251  0.91621024  1.6332111 -0.54837584 -1.1930381
## 1252  0.78784876  0.4637771 -0.72690896 -0.4131289
## 1253 -0.52990891 -0.5976884  0.01582429 -0.2948224
str(exrates.r)    #Shows there are 4 columns with 1252 instances
##  num [1:1252, 1:4] 0.186 0.592 0.147 0.792 -1.212 ...
##  - attr(*, "dimnames")=List of 2
##   ..$ : chr [1:1252] "2" "3" "4" "5" ...
##   ..$ : chr [1:4] "USD.EUR" "USD.GBP" "USD.CNY" "USD.JPY"
#creating a matrix for the volatility of the exchange markets
size <- na.omit(abs(exrates.r))
head(size)        #viewing the first 6 rows of the size matrix
##     USD.EUR    USD.GBP    USD.CNY    USD.JPY
## 2 0.1855770 0.41352605 0.03052233 0.08821260
## 3 0.5915427 0.26629486 0.08837968 0.44028690
## 4 0.1473405 0.39811737 0.02894123 0.25228994
## 5 0.7919091 0.70886373 0.12695761 1.37092779
## 6 1.2124033 0.04447127 0.09792040 0.03241316
## 7 0.3100091 0.54159233 0.05456676 0.82836254
#Creating a vector of names for the matrix that has the same name as
#the exrates matrix, but with ".size" appended at the end
colnames(size) <- paste(colnames(size), ".size", sep = "")
#viewing the first six rows exchange rates size matrix
head(size)
##   USD.EUR.size USD.GBP.size USD.CNY.size USD.JPY.size
## 2    0.1855770   0.41352605   0.03052233   0.08821260
## 3    0.5915427   0.26629486   0.08837968   0.44028690
## 4    0.1473405   0.39811737   0.02894123   0.25228994
## 5    0.7919091   0.70886373   0.12695761   1.37092779
## 6    1.2124033   0.04447127   0.09792040   0.03241316
## 7    0.3100091   0.54159233   0.05456676   0.82836254
#Creating an empty matrix with the dimensions of exrates.r
direction <- exrates.r
direction[exrates.r > 0] <- 1     #setting the matrix > 0 = 1
direction[exrates.r < 0] <- -1    #setting the matrix > 0 = -1
direction[exrates.r == 0] <- 0    #setting the matrix == 0 = 0
#setting the column names to the exchange market with ".dir" appended
colnames(direction) <- paste(colnames(exrates.r), ".dir", sep = "")

#Converting into a time series object
#Vector of only dates
#removing the first index of the dates because of the diff function
dates <- exrates$DATE[-1]
#Creating a matrix that has has the percent change, absolute value
#of the percent change, and the whether the foreign exchange 
#appreciated, depreciated, or stayed the same
values <- cbind(exrates.r, size, direction)
#Creating the data frame with the dates, returns, size, and direction
exrates.df = data.frame(dates = dates, returns = exrates.r,
    size = size, direction = direction)
#Viewing structure of the data to ensure all is looking normal
str(exrates.df)
## 'data.frame':    1252 obs. of  13 variables:
##  $ dates                : Date, format: "2013-01-29" "2013-01-30" ...
##  $ returns.USD.EUR      : num  0.186 0.592 0.147 0.792 -1.212 ...
##  $ returns.USD.GBP      : num  0.4135 0.2663 0.3981 -0.7089 -0.0445 ...
##  $ returns.USD.CNY      : num  0.0305 -0.0884 -0.0289 0.127 0.0979 ...
##  $ returns.USD.JPY      : num  -0.0882 0.4403 0.2523 1.3709 0.0324 ...
##  $ size.USD.EUR.size    : num  0.186 0.592 0.147 0.792 1.212 ...
##  $ size.USD.GBP.size    : num  0.4135 0.2663 0.3981 0.7089 0.0445 ...
##  $ size.USD.CNY.size    : num  0.0305 0.0884 0.0289 0.127 0.0979 ...
##  $ size.USD.JPY.size    : num  0.0882 0.4403 0.2523 1.3709 0.0324 ...
##  $ direction.USD.EUR.dir: num  1 1 1 1 -1 1 -1 -1 -1 1 ...
##  $ direction.USD.GBP.dir: num  1 1 1 -1 -1 -1 1 1 1 -1 ...
##  $ direction.USD.CNY.dir: num  1 -1 -1 1 1 -1 1 1 0 0 ...
##  $ direction.USD.JPY.dir: num  -1 1 1 1 1 1 1 -1 -1 1 ...
#Converts the matrix into a time series object
exrates.xts  <- na.omit(as.xts(values, dates))
#Viewing the structure of the time series to make sure nothing is wrong
str(exrates.xts)
## An 'xts' object on 2013-01-29/2018-01-26 containing:
##   Data: num [1:1252, 1:12] 0.186 0.592 0.147 0.792 -1.212 ...
##  - attr(*, "dimnames")=List of 2
##   ..$ : NULL
##   ..$ : chr [1:12] "USD.EUR" "USD.GBP" "USD.CNY" "USD.JPY" ...
##   Indexed by objects of class: [Date] TZ: UTC
##   xts Attributes:  
##  NULL
#converting from the xts object to a zooreg object
exrates.zr <- na.omit(as.zooreg(exrates.xts))
#Looking at the structure of the of the zooreg object
str(exrates.zr)
## 'zooreg' series from 2013-01-29 to 2018-01-26
##   Data: num [1:1252, 1:12] 0.186 0.592 0.147 0.792 -1.212 ...
##  - attr(*, "dimnames")=List of 2
##   ..$ : NULL
##   ..$ : chr [1:12] "USD.EUR" "USD.GBP" "USD.CNY" "USD.JPY" ...
##   Index:  Date[1:1252], format: "2013-01-29" "2013-01-30" "2013-01-31" "2013-02-01" "2013-02-04" ...
##   Frequency: 1

The result of the above code converts the daily foreign exchange rates to the percent change in the foreign exchange rates, creates a matrix of the absolute value of the percent change, and creates another matrix that indicates whether the exchange rate appreciated, depreciated, or remained the same. Finally, the above block combines all the data created and converts them in to a zoo time series object using the date vector.

Next, we will create an interactive visual plot of the percent daily change for the four exchange markets ranging from 2013 to 2018 using the ggplot2 and plotly libraries.

#Creates the title for the the plot
title.chg <- "Exchange Rates Percent Change"
#Creating the ggplot object for the percent change with a y-limit of -5% to 5%
#for each of the four foreign exchange market
p1 <- autoplot.zoo(exrates.zr[,1:4]) + ggtitle(title.chg) + ylim(-5,5)
#Displays an interactive plot for the ggplot object created above
ggplotly(p1)
#Sets a title for the size plot
title.size <- "Absolute Value of Percent Change"
#Creates a ggplot object for the size of the the change with the y limit 
#from 0 to 5
p2 <- autoplot.zoo(exrates.zr[,5:8]) + ggtitle(title.size) + ylim(0,5)

The above interactive plots show the daily percent change in the foriegn exchange markets relative to the U.S Dollar. These plots also show that the foreign exchange markets for the Japanese yen consistently displays the most volatile behavior, while the exchange market for the Chinese Yuan appears to be the most steady. The foreign exchange markets for the british pound and the euro have a fairly similar volatility, somewhere between the Yen and the Yuan. This makes intuitive sense as until recently, the UK was a part of the European Union and had their markets fairly entwined. There is some excess volatility in the GBP exchange market around June 2016 as the United Kingdom voted to leave the European Union, allowing for political speculation of the future of the UK.

The relatively flat line for the percent change in the exchange rate from USD to CNY shows that the value of the yuan is generally fixed against the price of the dollar, meaning it you could buy approximately the same amount of yuan with the U.S Dollar during any time in that timespan. This fixed price against the dollar is significant, because it keeps the Yuan cheap, even as the Chinese economy continues to grow.

Using this plot, we can see that the volatility in our earnings due to the foreign exchange market is primarily caused by our business in Japan. The volatility in GBP and Euro markets is not as significant as the Yen, but could cause significant issues to our cash flow, especially depending on the volume of revenue. If we were looking at potentially accessing the Chinese market to stabilize some of the volatility in our cash flow, we would want to do more research on the barrier of entry to the Chinese market as the economy of China is dominated by State owned enterprises, and thus the political risk may outweigh the financial gains. It would also be beneficial to see if our prices would be competitive in the Chinese market as the Chinese economy has been growing at a higher rate than the United States, while valuing the currency at approximately the same price, seemingly devaluing the Yuan.

Question 2: Foreign Exchange Market Relationships and Descriptive Statistics

Next, we will be interested in seeing how the foreign exchange markets interact, and whether events in the past have an effect on the current market. This will be viewed in terms of the autocorrelations in percent change and autocorrelations in the size of the change. Using the partial autocorrelation, we will be able to see how long of a memory the markets have and which days are significant in determining today’s market. First we will create an autocorrelation matrix for the percent change in the markets.

#Creating an autocorrelation matrix for each market
#The diagonal is shows the memory in the past for the same market
#The x-axis is the amount of lag 
#If x-axis is negative, the exchange market that is listed first
#in the title is the exchange market that is being lagged
acf(coredata(exrates.xts[,1:4]))

The vertical lines that appear along the x-axis show the autocorrelation coefficient, and vertical lines that appear above the dashed line have a significant autocorrelation coefficient. The autocorrelation matrix shows that there is some correlation in the percent change between the markets on the same day (zero lag), but not really any memory in the markets to create accurate forecasts with this data alone.

The most significant same day correlations are between the Euro/GBP interaction and between the Euro/JPY interaction. The interactions of the Chinese Yuan and every other market appears to be less significant than the all other interactions. There also appears to be little interaction between the change in the GBP exchange rate and the change in the JPY exchange rate.

Next, we will create an autocorrelation matrix for the sizes in the percent changes. Interactions that are significant will mean that a large magnitude of returns breeds more large magnitude of returns (independent of direction), and that a small magnitude of returns brees more smaller magnitude of returns.

#Creating an autocorrelation matrix for the foreign
#exchange markets for columns 5-7 (Size of the 
#percent change). Not including the japanese market
#to have larger plots
acf(coredata(exrates.xts[,5:7]))

Using the autocorrelation matrix for the magnitude of percent change in the foreign exchange markets, we can see that there is some memory within the same market. This can be determined by looking along the diagonal of the matrix and seeing a number of vertical lines outside the range of the confidence interval. The memory within the same market is also more relevant in the short-term (lags 1-5) as the frequency of autocorrelation intervals above the confidence interval is greater than the frequenct of significant autocorrelation terms with a greater lag; but, there seems to be some persistence in the market as there are some coefficients outside the confidence interval around a lag of 20. This graphic shows that it is likely that a certain magnitude of returns today will bring a like magnitude of returns in the future. Whether the exchange rate will appreciate or depreciate is not within the scope of this graphic as we are only looking at the size of the percent change.

Next we will use the partial autocorrelation between the foreign exchange markets to see if there is any correlation in the interactions between markets in the past, independent of what has happened between the lag date and the present.

#Creating a partial 
pacf(coredata(exrates.xts[,1:4]))

The partial autocorrelation matrix for the foreign exchange markets shows that there is a significant amount of memory in the GBP and EUR exchange markets in terms of the CNY exchange rate. This is particularly apparent around the 10 lag mark. The partial autocorrelation matrix also shows that there is a significant amount of memory in the JPY exhange in terms of past events in the CNY exchange. While it is interesting to note that the partial autocorrelation of the CNY exchange is significant with all the other exchanges, it could also be worth noticing that there is not a lot of memory in the CNY exchange when when looking at last event in its own exchange, which is typical for the other exchanges, but the CNY exchange does not appear to behave in the same manner as the other exchanges. This could be the result of the CNY exchange essentially mimicing the daily percent changes of the USD.

Next we will create a partial autocorrelation plot that will show whether the size of the percent change in the exchange rate in the past has an effect on the percent change of today’s rate, independent of what has happened between the lag and the present.

pacf(coredata(exrates.xts[,5:7]))

Comment on the pacf

data_moments <- function(data) {
    library(moments)                            #Package for skewness and kurtosis
    library(matrixStats)                        #Package for statistics on matrix columns
    mean.r <- colMeans(data)                    #Calculates the mean for each column
    median.r <- colMedians(data)                #Calculates the median for each column
    sd.r <- colSds(data)                        #Standard deviation for each column
    IQR.r <- colIQRs(data)                      #Difference between Q1 and Q3
    skewness.r <- skewness(data)                #Skewness for each column
    kurtosis.r <- kurtosis(data)                #kurtosis for each column
    #Creates a data frame with the statistics for each column
    result <- data.frame(mean = mean.r,
        median = median.r, std_dev = sd.r, 
        IQR = IQR.r, skewness = skewness.r, 
        kurtosis = kurtosis.r)
    return(result)
}
answer <- data_moments(exrates.xts[,5:8])
knitr::kable(answer, digits = 4)
mean median std_dev IQR skewness kurtosis
USD.EUR.size 0.4003 0.2935 0.3695 0.4313 1.7944 8.0424
USD.GBP.size 0.4008 0.2995 0.4266 0.4173 6.0881 93.5604
USD.CNY.size 0.1027 0.0601 0.1375 0.1154 3.9004 31.0222
USD.JPY.size 0.4533 0.3250 0.4455 0.4684 2.2201 10.4898